When should I remove the password from my Shopify store?

Once you have removed your password to allow your domain to be verified in Google Search Console, you can then leave your password off your store.

If you add your password again, it will interfere will Google Search Console crawling your website.

Although you won't yet be ready to take orders or speak to customers, you will be making it clear initially that your products are only for demonstration purposes.

Also, since you won't yet be running any ads and you haven't allowed enough time for your website to appear in organic search results, there will be very few visitors to your site until you're ready anyway.